Final answer:
To prove that (A ∪ B) - C equals (A - C) ∪ (B - C), we demonstrate that each element of one set is an element of the other set, showing the two sets are equal.
Step-by-step explanation:
To prove that (A ∪ B) - C = (A - C) ∪ (B - C) using the algebra of sets, we need to show that each side of the equation is a subset of the other.
First, let's take any element x in (A ∪ B) - C. By definition of set difference, this means x is in A ∪ B and not in C. Since x is in A ∪ B, x is in A or in B (or both). If x is in A and not in C, then x is in A - C. If x is in B and not in C, then x is in B - C. Therefore, x belongs to the set (A - C) ∪ (B - C).
Conversely, take any element x in (A - C) ∪ (B - C). This means that x is either in A - C or in B - C (or both). If x is in A - C, then x is in A and not in C; similarly, if x is in B - C, then x is in B and not in C. Hence, x is in A or B (or both) and not in C, which means x is in A ∪ B and not in C. Thus, x belongs to (A ∪ B) - C.
Since every element of (A ∪ B) - C is in (A - C) ∪ (B - C) and every element of (A - C) ∪ (B - C) is also in (A ∪ B) - C, the two sets are equal, which proves the original statement.